Stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flow

Abstract

We consider the stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flow problems. For elements of degree 4 or higher, we construct a right-inverse of the divergence operator that is stable uniformly in the polynomial degree $N$ from $L^p$ to ${\mathbf{W}}^{1,p}$, show that the associated inf-sup constant is bounded below by a constant that decays at worst like $N^{-3|\frac{1}{2}-\frac{1}{p}|}$, and construct local Fortin operators with stability constants explicit in the polynomial degree. We demonstrate these results with several numerical examples suggesting that the $p$-version method can offer superior convergence rates over the $h$-version method even in the non-Newtonian setting.